He was arrogant, dismissive of others work.
He was arrogant, a dismissive person allround.
Does 1. require an and conjunction as a rule or is ok as written? The second example sounds fine with or without one.
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Depends what you want to convey, if his dismissiveness is a symptom of his arrogance I'd have said "He was arrogant; dismissive of others' work." if he's arrogant and also dismissive then you are listing individual faults in which case you do need an and. I'd phrase the second sentence as "He was arrogant, an all around dismissive person." but as far as I know that's just a personal preference for particular sentence structure rather than driven by any formal rule I was ever taught.